The present invention relates to a method for determining the fluid flow permeability of porous media. In particular, the present invention relates to measuring the permeability by nuclear magnetic resonance (NMR) using the T.sub.2 relaxation time for the decay of the transverse magnetization of fluids saturating the medium under investigation.
The properties of fluids in porous media are of great relevance in many fields of science and engineering. There are numerous measurements which bear some importance on characterizing fluid properties in confined geometries like sandstone rocks. We list here a few: porosity, fluid flow permeability (both dc and ac), electrical conductivity, wettability, etc. Quantities like porosity and fluid flow permeability in porous rocks are of great relevance for determining, the producibility of petroleum reservoirs.
For porous media it has become customary to speak of the solid material which forms the "backbone" as the matrix and its complement as the pore space. Porosity is defined as the ratio of pore space volume inside the porous material to the total volume of the porous medium. Permeability is a measure for the ability of porous materials like e.g. porous rocks to permit fluid flow through the pore space. It generally increases with porosity, but also depends on other parameters of the rocks as e.g. the specific surface area of the pore space, the pore size distribution and the pore shape. The fluid flow permeability can vary by about 8 orders of magnitude in loose sediments and sedimentary rocks. It has the dimension of area and is defined by d'Arcy's law which relates the rate of fluid flow to the pressure differential between two parallel planes for inflow and outflow. The fluid flow permeability is measured in the laboratory by fitting sleeves to core samples which are often cylindrically shaped. The top and bottom of the core samples are connected to fluid inlets and outlets and a known pressure difference is applied across the sample. The fluid flow rate is measured for a set of different pressure gradients. Liquids or gases can be used as flowing medium, although the measurement using a liquid is generally easier as in most cases the liquid can be considered incompressible. The laboratory procedure therefore requires first to drill core plugs from core samples, which have to be cleaned with various solvents. In contrast the method of the present invention can be carried out with a nuclear magnetic resonance logging tool to measure in situ the transverse relaxation time of the fluids saturating an earth formation to accurately predict the fluid flow permeability of the earth formation.
Nuclear magnetic resonance (NMR) has been employed for some time to study fluids permeating the pore space of porous media [see J. R. Banavar and L. M. Schwartz. "Molecular Dynamics in Restricted Geometries", chapter 10, edited by J. Klafter and J. M. Drake, J. Wiley (1989)]. The fluid supplies the probe particles which diffuse in the pore space. Since the classic paper by Brownstein and Tarr (BT) [see K. R. Brownstein and C. E. Tarr, Physical Review A. 19, 2446 (1979)] it has been realized that nuclear spin relaxation can provide information about the pore space geometry. BT discussed the case of T.sub.1 and T.sub.2 relaxation in an isolated pore where the nuclear spins are relaxed by collisions with the pore walls. The interpretation of T.sub.1 measurements with this model for fluids in porous media can present several problems. It is assumed that the relaxation of the bulk fluid in the pore space is negligible in comparison with the surface relaxation. In the limit where the nuclear spins diffuse at a fast rate to the pore surface and the surface relaxation is in comparison relatively slow, the averaged relaxation curve can be related to the pore size probability distribution. In this so called fast diffusion limit where the lowest order relaxation mode dominates one still has to assume that the surface relaxation strength is uniform and the pores are isolated to relate the distribution of relaxation times uniquely to the pore size distribution. It is conceivable to have porous samples with the same pore size geometry but different levels of paramagnetic impurities which influence the surface relaxation velocity while the fluid flow permeability would remain unchanged. To obtain a reliable estimate of the fluid flow permeability with NMR one therefore has to perform an experiment which directly probes fluid transport in the porous medium like for example the diffusion of fluid molecules in the pore space. For T.sub.1 measurements the nuclear spin relaxation depends on the rate at which magnetization is carried to the surface but also on the surface relaxation velocity .rho.. As the surface relaxation strength .rho. has no bearing on permeability one can therefore hope to correlate T.sub.1 and the fluid flow permeability only for classes of materials with similar surface relaxation properties.
There is an increasing interest in applying NMR in well-bore environments to determine the properties of fluid carrying earth formations [see P. N. Sen. C. Straley, W. E. Kenyon and M. S. Whittingham, Geophysics, 55, 61-69(1990)]. This interest has been spurred by the introduction of a new generation of NMR logging tools by NUMAR [see [M. N. Miller, A. Paltiel. M. E. Gillen, J. Granot and J. C. Brouton, Society of Petroleum Engineers, SPE 20561, 321(1990)], which are already being used in the field. The new NMR logging tools are very well fitted to carry out the physical measurements required for our method of invention.
In the present invention, a measurement of the transverse relaxation time T.sub.2 for fluids in porous media is used to determine the permeability of the medium by taking advantage of magnetic field inhomogeneities across pores. For strong magnetic fields and in the fast diffusion limit the relaxation is determined to first order by the transport of magnetization through the pore space and not the surface relaxation velocity. It will be shown that it is possible to correlate T.sub.2 to a length characteristic of the pore space geometry which can also be determined independently from mercury injection experiments and thereby relate T.sub.2 to the fluid flow permeability. It is also feasible to study the degree to which the diffusion of fluid molecules is restricted by the pore space geometry. T.sub.2 for fluids in porous media is in general orders of magnitude shorter than T.sub.1 in marked contrast to the situation for bulk fluids. The main mechanism for T.sub.2 relaxation of the fluid spins in strong magnetic fields is due to the internal random magnetic field gradients generated by the difference in magnetic susceptibility for the fluid filling the pore space and the material making up the matrix of the porous medium. The spatial dependence of the internal gradients is determined by the pore space geometry and pore size distribution. The internal gradients in turn determine the rate at which the spins diffusing through the pore space loose their phase memory. The loss of phase memory can be monitored with a multi spin-echo pulse sequence like the Carr-Purcell-Meiboom-Gill (CPMG) pulse sequence [see S. Meiboom and D. Gill, Rev. Sci. Instr., 29, 688 (1958)].
The phenomenon of spin echoes essential to the present invention was first discovered in NMR by Erwin Hahn. In an inhomogeneous magnetic field nuclear spins will precess at a Larmor frequency, .nu..sub.L, determined by the local field. After an initial radiofrequency pulse which tips the spins into a plane transverse to the direction of the applied static magnetic field the spins are all in phase and the sum of the total transverse magnetization is at the maximum possible value. Due to the spread in precession frequencies the spins will dephase and the macroscopic magnetization measured with the NMR instrument will decay. It is useful to remember here that the macroscopic magnetization is a vector sum of the magnetic moments of the spins which vanishes when the phases of the magnetic moments are random. One can reverse the dephasing process by applying a 180 degree pulse a time .tau./2 after the initial radio-frequency pulse which tipped the nuclear spins into the transverse plane. Immediately after this pulse a spin which precesses at a faster frequency than the average lags behind by an angle which is exactly the same angle by which it was ahead of the average immediately before the 180 degree pulse. Similarly spins precessing at a frequency slower than the average are now ahead. A time .tau./2 after the 180 degree pulse the spins will be again be in phase and one can observe a spin-echo. Spins diffusing will be subject to different local fields between the time the first pulse was applied and the detection of the spin echo. As their Larmor frequency is not constant the refocusing of magnetization will be incomplete and the echo will be attenuated. The degree of attenuation depends on the displacement and field inhomogeneity. This attenuation can be used to measure diffusion constants in fluids and to probe the diffusion of fluid spins in the pore space of porous media.